Elementary schematic geometry

TL;DR

This paper introduces a functorial formalism using geometric contexts and elementary schemes to unify the study of various geometric objects through local properties and base change stability.

Contribution

It develops a categorical framework for geometry based on Grothendieck topologies and elementary schemes, unifying diverse geometric theories.

Findings

Provides a formal language for geometric contexts

Shows geometry as local properties stable under base change

Unifies different geometric categories through functorial formalism

Abstract

This paper exposes the language of geometric contexts and elementary schemes, which is a functorial formalism to study categories of geometric objects such as schemes, topological manifolds, differential manifolds, analytic manifolds, etc. Through the theory of Grothendieck topologies and geometric contexts, geometry turns out to be the study of local properties which are stable under base change.